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\title{复变函数测验3}
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\date{2024 年 5 月 20 日}
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\begin{enumerate}

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\item  %Problem 1 第75页例子3.5
设积分路径 $C$ 是连接点 $1$ 和点 $i$ 的直线段，使用参数方程法，计算积分 $$\int_C (x+y+ix^2+iy^2)dz. $$

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\item  %Problem 2 第83页例子3.8
求出 $f(z)=z\sin(z)$ 的原函数，并使用原函数计算积分 
$$\int_{-\pi i}^{\pi i} z\sin (z) dz. $$

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\item  %Problem 3 第86页例子3.12
设 $C$ 为包含圆周 $|z|=5$ 在内的正向简单闭曲线，使用复周线上的柯西积分定理，
计算积分 $$\int_C \frac{2z+3}{z^2+3z}dz. $$


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\item  %Problem 4 第88页例子3.15 
设 $C$ 为圆周 $| z+2i |=1$. 使用柯西积分公式，计算 $$\int_C \frac{dz}{z^2(z^2+4)}.$$

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\item  %Problem 5 第91页例子3.17
使用解析函数的导函数的柯西积分公式，计算积分 $$\int_{|z|=2} \frac{\sin(z)dz}{(z+i)^4}. $$


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\item  %Problem 6 103页习题12
设函数 $f(z)$ 定义如下，使用柯西积分公式，计算 $f(i)$ 与 $f'(i)$.  
 $$f(z) = \int_{|z|=2} \frac{\zeta^2+3\zeta+4}{\zeta -z}d\zeta. $$ 

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\item  %Problem 7 第97页例子3.21 
验证 $u=\frac{y}{x^2+y^2}$ 是调和函数，并求以 $u$ 为实部的解析函数 $f(z)$ 使得 $f(1+i)=1-i$. 

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\item  %Problem 8 第106页习题15
%设函数 $f(z)$ 在区域 $D$ 内解析，证明 
%$$\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 4 \frac{\partial^2 f(z)}{\partial z \partial \bar{z}}. $$
设 $f(z)=u+iv$ 是解析函数，设 $u+v=(x-y)(x^2+4xy+y^2)-2(x+y)$. 求 $f(z)$. 


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\end{enumerate}


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